Reverse Fatou lemma (${\mathrm{limsup}}_n\mathbb{E}{Y}_n\le \mathbb{E}{\mathrm{limsup}}_n{Y}_n$)

If $Y_n$ are random variables bounded above by an integrable function $Z$, with all relevant expectations well defined, Fatou's lemma applied to the nonnegative variables $Z-Y_n$ gives ${\mathrm{limsup}}_n\mathbb{E}{Y}_n\le \mathbb{E}{\mathrm{limsup}}_n{Y}_n$. In particular, bounded indicator random variables that tend almost surely to zero have probabilities tending to zero.